2/10/2005 CHEM 408 - Sp05L5 - 1......)3,2,1(321χχχ=Ψ)}1(or )1({)1()1(βαψχ×=ii)1()1(∑=νννϕψiiC• adopt orbital approximation for electronic Ψ:multi-eΨelSlater det. of1e spin-orbitalsspin-orbitals = spatial MO × spin fn.• write MOs as linear combinations of basis functions (AOs) ϕν: with• for given basis set {ϕi}, find optimum 1e MOs {ψi} by minimizing......)3,2,1(...)3,2,1(.........)3,2,1(ˆ...)3,2,1(...ˆ321321ττττττddddddHHelappΨΨΨΨ=∫∫∫∫∫∫∗∗Ψ• difficult to solve; requires iteration to “self-consistency”• each electron interacts with average charge distribution “field” created by the remaining electronsHFHF--SCF CalculationsSCF Calculations2/10/2005 CHEM 408 - Sp05L5 - 2• consider the sort of terms that arise in using a simple 2-e Ψ: appHΨˆ{})2()1()2()1()2()1(21)2,1(3ααψψψψααψψabbaba−==Ψ{}{}{})2()1()2()1()2,1(ˆ)2()1()2()1( )2()2()1()1(21ˆ2121abbaabbaHrdrddddHψψψψψψψψααωααωτ−−×=ΨΨ∗∗∗∗∫∫∫∫∫rr11{})2()1(ˆ)2()1( )2()1(ˆ)2()1(- )2()1(ˆ)2()1(- )2()1(ˆ)2()1( 2121ababbaababbababaHHHHrdrdψψψψψψψψψψψψψψψψ∗∗∗∗∗∗∗∗+=∫∫rr• evaluating: spin parts separate and don’t contribueAABBe labels don’t matter so only 2 distinct terms A & B)2()1(ˆ)2()1()2()1(ˆ)2()1( 2121 abbababaHrdrdHrdrdψψψψψψψψ∗∗∗∗∫∫∫∫−=rrrr2/10/2005 CHEM 408 - Sp05L5 - 3• recall what’s in Ĥ:)2,1()2(ˆ)1(ˆ)2,1(ˆeeNNVVhhH +++=∑−−∇−=JJJRrZh nuclei121||21)1(ˆrr||121rrVeerr−+=∑>−+=KJKJKJNNRRZZV nuclei||rr1e or core ĥNN repulsion ( constant)ee repulsion(difficult)bbaaHHdhh +=Ψ+Ψ∫τ)}2(ˆ)1(ˆ{ *)2()2()1()1(ˆ)1()2()2()1()1(ˆ)1()2()1()1(ˆ)2()1()2()1()1(ˆ)2()1()1(ˆ 21212121*bababbaaabbababardhrdrdhrdhrdrdhrdrddhψψψψψψψψψψψψψψψψτ∗∗∗∗∗∗∗∗∫∫∫∫∫∫∫∫∫−=−=ΨΨrrrrrrrr10NNNNVdV =ΨΨ∫τ* • VNNterm is trivial:(because VNNdoesn’t depend on e coordinates)• ĥ terms reduce to 1e integrals:∫∗≡ )1()1(ˆ)1(1 iiiihrdHψψrwhere2/10/2005 CHEM 408 - Sp05L5 - 4• note that if analogous singlet is considered,the exchange term Kabterm disappears: ababeeKJdV −=ΨΨ∫ *τ• Veeterms don’t simplify:|||)2(||)1(|)2()1(||1)2()1(2122212121rrrdrdrrrdrdJbababaabrrrrrrrr−=−≡∫∫∫∫∗∗ψψψψψψ)2()1(||1)2()1(2121 abbaabrrrdrdKψψψψrrrr−=∗∗∫∫where:“Coulomb integral”classical interaction of charge distributions |ψa|2& |ψb|2“exchange integral”purely QM effect of exchange symmetry requirements on Ψ• in summary, for ααψψba=Ψ )2,1(3NNababbbaaVKJHHdH +−++=ΨΨ∫τ3*3ˆβαψψba=Ψ )2,1(1NNabbbaaVJHHdH +++=ΨΨ∫τ1*1ˆ2/10/2005 CHEM 408 - Sp05L5 - 5More generally, for a closed-shell system, one consisting of 2nelectrons electrons filling n spatial MOs in α, βpairs,NNninjijijiiVKJHdH +⎭⎬⎫⎩⎨⎧−+=ΨΨ∑∑∫==∗11)2(2ˆτ22221...)2...3,2,1(nnψψψ=Ψ)1()2()1()1(1121βψαψψ=where etc.:∫∑⎭⎬⎫⎩⎨⎧−−∇−≡∗)1(||)1( nuclei121211 iJJJiiiRrZrdHψψrrr)2()1(||1)2()1(2121 jijiijrrrdrdJψψψψrrrr−≡∗∗∫∫)2()1(||1)2()1(2121 ijjiijrrrdrdKψψψψrrrr−=∗∗∫∫∑>−+=KJKJKJNNRRZZV nuclei||rrwherecoreCoulombexchangenuclear repulsion2/10/2005 CHEM 408 - Sp05L5 - 6• now write the MOs in terms of basis functions {ϕµ} µ = 1…K:∑==KiiC1µµµϕψ(MOs i, j, …; basis fncs. or “AOs”µ, ν, …) • inserting this expansion into the preceding equations and minimizing <Ĥ>Ψwith respect to the aµias in the HMO theory (i.e. requiring for all µ, i yields the Roothan-Hall eqns:0/ˆ=∂><∂iCHµµ = 1…K)}|()|{(2111νσµλλσµνλσλσµνµν−+≡∑∑==KKcorePHF)1(||)1( nuclei121211νµµνϕϕ⎭⎬⎫⎩⎨⎧−−∇−≡∑∫∗JJJcoreRrZrdHrrr)1()1(1∗∗∫≡νµµνϕϕrdSr)2()2(1)1()1()|(1221σλνµϕϕϕϕλσµν∗∗∫∫=rrdrdrrset of K linear eqnsF = Fock matrixS = overlap matrixHcore= 1e core matrix2e Coulomb (µν|λσ) and exchange (µλ|νσ) integrals∑=∗≡niiiCCP1 MO2σλλσP density matrix0)(1=−∑=KiiCSFννµνµνε2/10/2005 CHEM 408 - Sp05L5 - 7• The Roothan-Hall eqns. are superficially like the secular eqns. from HMO theory. Using comparable notation:0)(=−ΣiiCSFνµνµννε1)1()1(ˆ)1(τϕϕνπµµνdhH∫∗≡0)(=−ΣiiCSHνµνµννε)}|()|{(2111νσµλλσµνλσλσµνµν−+≡∑∑==KKcorePHFHMO theoryHF-SCF theory• but the HF problem is not really a linear problem -- the Fockmatrix depends on the target coefficients {Cνi} through the density matrix P• whereas the HMO theory can be solved directly, the Roothan-Hall eqns. must be solved iteratively ESCHC =ESCFC=(or)2/10/2005 CHEM 408 - Sp05L5 - 8construct F(n)from P(n-1)& solve FC=SCE for C(n)& E(n)construct new P(n)from C(n)self-consistency achieved: C (or P) & E give best estimates of Ψi& Eiguess initial density matrix P(0)calc. & store* integralscoreHSµνµν ,)|(λσµνyesnoP(n) = P(n-1)?P(n-1)← P(n)The HF SCF Procedure2/10/2005 CHEM 408 - Sp05L5 - 92 Electron Integrals:)2()2(1)1()1()|(1221σλνµϕϕϕϕλσµν∗∗∫∫=rrdrdrr• distributed over 4 centers, e.g.r12e1e2ABCD||||21||||21DCBA2211||1 )s1s1|s1s1(DCBARrRrRrRreerreerdrdrrrrrrrrrrrr−−−−−−−−−∝∫∫Figs. from Szabo & Ostland, Modern Quantum Chemistry (MacMillan, 1982)• why Gaussian basis functions?STO-nG funcs.2/10/2005 CHEM 408 - Sp05L5 - 10• # integrals = (#basis fns)4/8- sets practical limits to HF calcs.⇒“Nb4scaling”Basis set: 6-31G** = 6-31G(d,p)C atom: 1s, 2s(1), 2s(2), 2p(1), 2p(2), 2d = 15H atom: 1s(1), 1s(2), 1p = 5Nb= 180; #I = 131,220,000 http://www.emsl.pnl.gov/forms/basisform.html• example: benzene• “tricks” reduce scaling to ~Nb2.7- symmetry reductions- integral size screening- fast multipole methods)exp(2rdiiiαϕ−Σ=αidi2/10/2005 CHEM 408 - Sp05L5 - 11Some Comments on HF Performance (≥pVDζbasis)1-5:• geometries typically well described1,2• bond lengths: heavy atoms X-Y ±0.03 Å; H-X ±0.015 Åsystematically too long• bond angles: ±1.5°• dihedral angles: also good• intermolecular complexes usually too loose, but H-bonded structures can be accurate• transition state structures are often reasonably described4• energies of protonation & deprotonation1,2 often accurate: • ionization potentials calculated via Koopman’s